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23
On the Solution of Traveling Salesman Problems
 DOC. MATH. J. DMV
, 1998
"... Following the theoretical studies of J.B. Robinson and H.W. Kuhn in the late 1940s and the early 1950s, G.B. Dantzig, R. Fulkerson, and S.M. Johnson demonstrated in 1954 that large instances of the TSP could be solved by linear programming. Their approach remains the only known tool for solving TS ..."
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Cited by 163 (7 self)
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Following the theoretical studies of J.B. Robinson and H.W. Kuhn in the late 1940s and the early 1950s, G.B. Dantzig, R. Fulkerson, and S.M. Johnson demonstrated in 1954 that large instances of the TSP could be solved by linear programming. Their approach remains the only known tool for solving TSP instances with more than several hundred cities; over the years, it has evolved further through the work of M. Grötschel , S. Hong , M. Jünger , P. Miliotis , D. Naddef , M. Padberg
CHVATAL CLOSURES FOR MIXED INTEGER PROGRAMMING PROBLEMS
, 1990
"... Chvátal introduced the idea of viewing cutting planes as a system for proving that every integral solution of a given set of linear inequalities satisfies another given linear inequality. This viewpoint has proven to be very useful in many studies of combinatorial and integer programming problems. T ..."
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Cited by 61 (0 self)
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Chvátal introduced the idea of viewing cutting planes as a system for proving that every integral solution of a given set of linear inequalities satisfies another given linear inequality. This viewpoint has proven to be very useful in many studies of combinatorial and integer programming problems. The basic ingredient in these cuttingplane proofs is that for a polyhedron P and integral vector w, if max(wx]x ~ P, wx integer} = t, then wx ~ t is valid for all integral vectors in P. We consider the variant of this step where the requirement that wx be integer may be replaced by the requirement that #x be integer for some other integral vector #. The cuttingplane proofs thus obtained may be seen either as an abstraction of Gomory's mixed integer cuttingplane technique or as a proof version of a simple class of the disjunctive cutting planes studied by Balas and Jeroslow. Our main result is that for a given polyhedron P, the set of vectors that satisfy every cutting plane for P with respect to a specified subset of integer variables is again a polyhedron. This allows us to obtain a finite recursive procedure for generating the mixed integer hull of a polyhedron, analogous to the process of repeatedly taking Chvátal closures in the integer programming case. These results are illustrated with a number of examples from combinatorial optimization. Our work can be seen as a continuation of that of Nemhauser and Wolsey on mixed integer cutting planes.
TSP cuts which do not conform to the template paradigm
 IN COMPUTATIONAL COMBINATORIAL OPTIMIZATION
, 2001
"... The first computer implementation of the DantzigFulkersonJohnson cuttingplane method for solving the traveling salesman problem, written by Martin, used subtour inequalities as well as cutting planes of Gomory’s type. The practice of looking for and using cuts that match prescribed templates in c ..."
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Cited by 25 (1 self)
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The first computer implementation of the DantzigFulkersonJohnson cuttingplane method for solving the traveling salesman problem, written by Martin, used subtour inequalities as well as cutting planes of Gomory’s type. The practice of looking for and using cuts that match prescribed templates in conjunction with Gomory cuts was continued in computer codes of Miliotis, Land, and Fleischmann. Grötschel, Padberg, and Hong advocated a different policy, where the template paradigm is the only source of cuts; furthermore, they argued for drawing the templates exclusively from the set of linear inequalities that induce facets of the TSP polytope. These policies were adopted in the work of Crowder and Padberg, in the work of Grötschel and Holland, and in the work of Padberg and Rinaldi; their computer codes produced the most impressive computational TSP successes of the nineteen eighties. Eventually, the template paradigm became the standard frame of reference for cutting planes in the TSP. The purpose of this paper is to describe a technique
WorstCase Comparison of Valid Inequalities for the TSP
 MATH. PROG
, 1995
"... We consider most of the known classes of valid inequalities for the graphical travelling salesman polyhedron and compute the worstcase improvement resulting from their addition to the subtour polyhedron. For example, we show that the comb inequalities cannot improve the subtour bound by a factor gr ..."
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Cited by 25 (1 self)
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We consider most of the known classes of valid inequalities for the graphical travelling salesman polyhedron and compute the worstcase improvement resulting from their addition to the subtour polyhedron. For example, we show that the comb inequalities cannot improve the subtour bound by a factor greater than 10/9. The corresponding factor for the class of clique tree inequalities is 8/7, while it is 4/3 for the path configuration inequalities.
The Circuit Polytope: Facets
, 1994
"... Given an undirected graph G = (V; E) and a cost vector c 2 IR E , the weighted girth problem is to find a circuit in G having minimum total cost. This problem is in general NPhard since the traveling salesman problem can be reduced to it. A promising approach to hard combinatorial optimization p ..."
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Cited by 15 (1 self)
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Given an undirected graph G = (V; E) and a cost vector c 2 IR E , the weighted girth problem is to find a circuit in G having minimum total cost. This problem is in general NPhard since the traveling salesman problem can be reduced to it. A promising approach to hard combinatorial optimization problems is given by the socalled cutting plane methods. These involve linear programming techniques based on a partial description of the convex hull of the incidence vectors of possible solutions. We consider the weighted girth problem in the case where G is the complete graph K n and study the facial structure of the circuit polytope P n C and some related polyhedra. In the appendix we give complete characterizations of P n C for n up to 6.
Progress in linear programmingbased algorithms for integer programming: An exposition
 INFORMS JOURNAL ON COMPUTING
, 2000
"... This paper is about modeling and solving mixed integer programming (MIP) problems. In the last decade, the use of mixed integer programming models has increased dramatically. Fifteen years ago, mainframe computers were required to solve problems with a hundred integer variables. Now it is possible t ..."
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Cited by 14 (0 self)
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This paper is about modeling and solving mixed integer programming (MIP) problems. In the last decade, the use of mixed integer programming models has increased dramatically. Fifteen years ago, mainframe computers were required to solve problems with a hundred integer variables. Now it is possible to solve problems with thousands of integer variables on a personal computer and obtain provably good approximate solutions to problems such as set partitioning with millions of binary variables. These advances have been made possible by developments in modeling, algorithms, software, and hardware. This paper focuses on effective modeling, preprocessing, and the methodologies of branchandcut and branchandprice, which are the techniques that make it possible to treat problems with either a very large number of constraints or a very large number of variables. We show how these techniques are useful
The symmetric traveling salesman polytope: New facets from the graphical relaxation
 MATHEMATICS OF OPERATIONS RESEARCH
, 2007
"... ..."
Restricted 2Factor Polytopes
, 1998
"... The optimal krestricted 2factor problem consists of finding, in a complete undirected graph K n , a minimum cost 2factor (subgraph having degree 2 at every node) with all components having more than k nodes. The problem is a relaxation of the wellknown symmetric travelling salesman problem, and ..."
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Cited by 8 (0 self)
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The optimal krestricted 2factor problem consists of finding, in a complete undirected graph K n , a minimum cost 2factor (subgraph having degree 2 at every node) with all components having more than k nodes. The problem is a relaxation of the wellknown symmetric travelling salesman problem, and is equivalent to it when n 2 k n \Gamma 1. We study the krestricted 2factor polytope. We present a large class of valid inequalities, called bipartition inequalities, and describe some of their properties; some of these results are new even for the travelling salesman polytope. For the case k = 3, the trianglefree 2factor polytope, we derive a necessary and sufficient condition for such inequalities to be facet inducing. Research partially supported by a grant from N.S.E.R.C. of Canada 1 Introduction A 2factor of an undirected graph G = (V; E) is a spanning subgraph H of G that has degree 2 at each node. Equivalently, it is a set of nodedisjoint circuits that include all of t...